Integrand size = 4, antiderivative size = 26 \[ \int \text {sech}^5(x) \, dx=\frac {3}{8} \arctan (\sinh (x))+\frac {3}{8} \text {sech}(x) \tanh (x)+\frac {1}{4} \text {sech}^3(x) \tanh (x) \]
Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \text {sech}^5(x) \, dx=\frac {3}{8} \arctan (\sinh (x))+\frac {3}{8} \text {sech}(x) \tanh (x)+\frac {1}{4} \text {sech}^3(x) \tanh (x) \]
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {3042, 4255, 3042, 4255, 3042, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \text {sech}^5(x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc \left (\frac {\pi }{2}+i x\right )^5dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {3}{4} \int \text {sech}^3(x)dx+\frac {1}{4} \tanh (x) \text {sech}^3(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{4} \int \csc \left (i x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {3}{4} \left (\frac {\int \text {sech}(x)dx}{2}+\frac {1}{2} \tanh (x) \text {sech}(x)\right )+\frac {1}{4} \tanh (x) \text {sech}^3(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{4} \left (\frac {1}{2} \tanh (x) \text {sech}(x)+\frac {1}{2} \int \csc \left (i x+\frac {\pi }{2}\right )dx\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {3}{4} \left (\frac {1}{2} \arctan (\sinh (x))+\frac {1}{2} \tanh (x) \text {sech}(x)\right )+\frac {1}{4} \tanh (x) \text {sech}^3(x)\) |
3.6.80.3.1 Defintions of rubi rules used
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.96 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81
method | result | size |
default | \(\left (\frac {\operatorname {sech}\left (x \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (x \right )}{8}\right ) \tanh \left (x \right )+\frac {3 \arctan \left ({\mathrm e}^{x}\right )}{4}\) | \(21\) |
parallelrisch | \(\frac {3 i \ln \left (i+\coth \left (x \right )-\operatorname {csch}\left (x \right )\right )}{8}-\frac {3 i \ln \left (-i+\coth \left (x \right )-\operatorname {csch}\left (x \right )\right )}{8}+\frac {3 \,\operatorname {sech}\left (x \right ) \tanh \left (x \right )}{8}+\frac {\operatorname {sech}\left (x \right )^{3} \tanh \left (x \right )}{4}\) | \(42\) |
risch | \(\frac {{\mathrm e}^{x} \left (3 \,{\mathrm e}^{6 x}+11 \,{\mathrm e}^{4 x}-11 \,{\mathrm e}^{2 x}-3\right )}{4 \left (1+{\mathrm e}^{2 x}\right )^{4}}+\frac {3 i \ln \left ({\mathrm e}^{x}+i\right )}{8}-\frac {3 i \ln \left ({\mathrm e}^{x}-i\right )}{8}\) | \(52\) |
Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (20) = 40\).
Time = 0.24 (sec) , antiderivative size = 461, normalized size of antiderivative = 17.73 \[ \int \text {sech}^5(x) \, dx=\frac {3 \, \cosh \left (x\right )^{7} + 21 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + 3 \, \sinh \left (x\right )^{7} + {\left (63 \, \cosh \left (x\right )^{2} + 11\right )} \sinh \left (x\right )^{5} + 11 \, \cosh \left (x\right )^{5} + 5 \, {\left (21 \, \cosh \left (x\right )^{3} + 11 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + {\left (105 \, \cosh \left (x\right )^{4} + 110 \, \cosh \left (x\right )^{2} - 11\right )} \sinh \left (x\right )^{3} - 11 \, \cosh \left (x\right )^{3} + {\left (63 \, \cosh \left (x\right )^{5} + 110 \, \cosh \left (x\right )^{3} - 33 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 3 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{6} + 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} + 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} + 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} + 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} + 3 \, \cosh \left (x\right )^{5} + 3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (21 \, \cosh \left (x\right )^{6} + 55 \, \cosh \left (x\right )^{4} - 33 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right ) - 3 \, \cosh \left (x\right )}{4 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{6} + 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} + 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} + 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} + 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} + 3 \, \cosh \left (x\right )^{5} + 3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]
1/4*(3*cosh(x)^7 + 21*cosh(x)*sinh(x)^6 + 3*sinh(x)^7 + (63*cosh(x)^2 + 11 )*sinh(x)^5 + 11*cosh(x)^5 + 5*(21*cosh(x)^3 + 11*cosh(x))*sinh(x)^4 + (10 5*cosh(x)^4 + 110*cosh(x)^2 - 11)*sinh(x)^3 - 11*cosh(x)^3 + (63*cosh(x)^5 + 110*cosh(x)^3 - 33*cosh(x))*sinh(x)^2 + 3*(cosh(x)^8 + 8*cosh(x)*sinh(x )^7 + sinh(x)^8 + 4*(7*cosh(x)^2 + 1)*sinh(x)^6 + 4*cosh(x)^6 + 8*(7*cosh( x)^3 + 3*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 + 30*cosh(x)^2 + 3)*sinh(x)^ 4 + 6*cosh(x)^4 + 8*(7*cosh(x)^5 + 10*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4 *(7*cosh(x)^6 + 15*cosh(x)^4 + 9*cosh(x)^2 + 1)*sinh(x)^2 + 4*cosh(x)^2 + 8*(cosh(x)^7 + 3*cosh(x)^5 + 3*cosh(x)^3 + cosh(x))*sinh(x) + 1)*arctan(co sh(x) + sinh(x)) + (21*cosh(x)^6 + 55*cosh(x)^4 - 33*cosh(x)^2 - 3)*sinh(x ) - 3*cosh(x))/(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x) ^2 + 1)*sinh(x)^6 + 4*cosh(x)^6 + 8*(7*cosh(x)^3 + 3*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 + 30*cosh(x)^2 + 3)*sinh(x)^4 + 6*cosh(x)^4 + 8*(7*cosh(x) ^5 + 10*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 + 15*cosh(x)^4 + 9*cosh(x)^2 + 1)*sinh(x)^2 + 4*cosh(x)^2 + 8*(cosh(x)^7 + 3*cosh(x)^5 + 3 *cosh(x)^3 + cosh(x))*sinh(x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (27) = 54\).
Time = 1.17 (sec) , antiderivative size = 422, normalized size of antiderivative = 16.23 \[ \int \text {sech}^5(x) \, dx=\frac {3 \tanh ^{8}{\left (\frac {x}{2} \right )} \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} - \frac {5 \tanh ^{7}{\left (\frac {x}{2} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} + \frac {12 \tanh ^{6}{\left (\frac {x}{2} \right )} \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} + \frac {3 \tanh ^{5}{\left (\frac {x}{2} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} + \frac {18 \tanh ^{4}{\left (\frac {x}{2} \right )} \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} - \frac {3 \tanh ^{3}{\left (\frac {x}{2} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} + \frac {12 \tanh ^{2}{\left (\frac {x}{2} \right )} \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} + \frac {5 \tanh {\left (\frac {x}{2} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} + \frac {3 \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} \]
3*tanh(x/2)**8*atan(tanh(x/2))/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh (x/2)**4 + 16*tanh(x/2)**2 + 4) - 5*tanh(x/2)**7/(4*tanh(x/2)**8 + 16*tanh (x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) + 12*tanh(x/2)**6*atan(t anh(x/2))/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/ 2)**2 + 4) + 3*tanh(x/2)**5/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/ 2)**4 + 16*tanh(x/2)**2 + 4) + 18*tanh(x/2)**4*atan(tanh(x/2))/(4*tanh(x/2 )**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) - 3*tanh(x /2)**3/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)* *2 + 4) + 12*tanh(x/2)**2*atan(tanh(x/2))/(4*tanh(x/2)**8 + 16*tanh(x/2)** 6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) + 5*tanh(x/2)/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) + 3*atan(tanh(x/ 2))/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \text {sech}^5(x) \, dx=\frac {3 \, e^{\left (-x\right )} + 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} - \frac {3}{4} \, \arctan \left (e^{\left (-x\right )}\right ) \]
1/4*(3*e^(-x) + 11*e^(-3*x) - 11*e^(-5*x) - 3*e^(-7*x))/(4*e^(-2*x) + 6*e^ (-4*x) + 4*e^(-6*x) + e^(-8*x) + 1) - 3/4*arctan(e^(-x))
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \text {sech}^5(x) \, dx=\frac {3}{16} \, \pi - \frac {3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 20 \, e^{\left (-x\right )} - 20 \, e^{x}}{4 \, {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{2}} + \frac {3}{8} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \]
3/16*pi - 1/4*(3*(e^(-x) - e^x)^3 + 20*e^(-x) - 20*e^x)/((e^(-x) - e^x)^2 + 4)^2 + 3/8*arctan(1/2*(e^(2*x) - 1)*e^(-x))
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \text {sech}^5(x) \, dx=\frac {3\,\mathrm {atan}\left ({\mathrm {e}}^x\right )}{4}+\frac {3\,\mathrm {sinh}\left (x\right )}{8\,{\mathrm {cosh}\left (x\right )}^2}+\frac {\mathrm {sinh}\left (x\right )}{4\,{\mathrm {cosh}\left (x\right )}^4} \]